Encouraged and emboldened (embiggened?) by the ingenuity displayed by some of our readers, I’d like to see what sort of response we get to this Problem of the Week:

Establish the following identity: for all natural numbers .

(Here we make the convention .) I find this problem tantalizing because it *looks* as if there should be some sort of conceptual proof — can you find one?

Please send your solutions to **topological[dot]musings[At]gmail[dot]com** by **Wednesday, June 11**, 11:59pm (UTC); do **not** submit solutions in Comments. Everyone with a correct solution gets entered in our Hall of Fame! We look forward to your response.

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June 10, 2008 at 8:15 am

JoeHmm, that’s the second time I’ve seen the word embiggened used in the past five minutes.

http://unapologetic.wordpress.com/2008/05/30/blath-captcha/

June 10, 2008 at 11:39 am

Todd TrimbleYeah, it’s from an episode of The Simpsons, where the legendary founder of Springfield, Jebediah Springfield, is quoted as having said, “A noble spirit embiggens the smallest man.” I acknowledge that John’s recent post

mayhave played a role here, although it’s not self-evident to me because the word immediately came to mind when I wrote “emboldened”. Anyway…June 10, 2008 at 5:07 pm

Paul ShearerI’m not getting anywhere with generating functions or the bijections suggested by the formula. I hope somebody comes up with a good bijective proof, the formula is just begging for it!

June 10, 2008 at 8:07 pm

Todd TrimbleIt is indeed! Watch this space though…

June 12, 2008 at 11:09 am

Solution to POW-4: A curious identity involving self-powers « Todd and Vishal’s blog[…] Week (POW), Puzzles by Todd Trimble Tags: combinatorics, trees The solutions are in! I thought last week’s problem might have been a little more challenging than problems of previous weeks — the identity is […]

June 12, 2008 at 3:42 pm

John ArmstrongWhy are you so surprised? It’s a perfectly cromulent word.